The interplay between effects of High-Energy QCD and Transverse-Momentum Dependent (TMD) Factorization has become a topic of intensive theoretical studies in recent years, see e.g. Refs. [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}. Regions of applicability of these two formalisms overlap when observables related with transverse-momentum (\(q_T\) ) are studied in the process containing another large scale \(M\)  in hadronic or lepton-hadron collision with large center-of-mass energy \(\sqrt{S}\) , while all three scales become hierarchical: \(\sqrt{S}\gg M \gg q_T\) . The rapidity-evolution equation for gluon TMD Parton Distribution Function has been conjectured few years ago [1]}, [2]}, which unifies the non-linear Balitsky-Kovchegov [13]}, [14]} evolution equation in the small-\(x\)  limit with the linear Collins-Soper-Sterman(CSS)-type evolution of TMD PDFs familiar at moderate values of \(x\)  (see e.g. the monograph [15]} for the review of TMD factorization). Unfortunately the systematic study of this evolution equation is a very challenging task. Its real-emission kernel is essentially three-dimensional, interpolating between small-\(x\)  (Regge) limit with characteristic non-trivial transverse-momentum dynamics and moderate-\(x\)  limit, for which longitudinal momentum conservation is crucial From this point of view, it is interesting to study the relation between real emission kernel of Refs. [1]}, [2]} and TMD gluon-gluon splitting function derived in Ref. [18]}. More recently, the origin of corrections \(\sim (\alpha _s \ln ^2 (M^2/q_T^2))^n\)  – the so-called “Sudakov” double logarithms, in the approach of Refs. [1]}, [2]} had been further clarified in the Ref. [3]} where the roles of conformally-invariant cut-off for rapidity divergences as well as of conservation of dominating light-cone component of momentum where emphasized. However, connection to the physics of standard Balitsky-Fadin-Kuraev-Lipatov (BFKL) [22]}, [23]}, [24]} evolution equation, which is the cornerstone of High-Energy QCD, seems to be lost in these recent developments.
